In this section, we derive the statistical error rate for estimation of the true parameter vector, \(\psi ^*\) , in a gamma frailty illness-death model with non-convex penalty, specifically in high-dimensional settings where \(d > n\)  with a sparsity level denoted \(\Vert \psi ^*\Vert _{0} = s\) . The result extends the statistical rate for penalized estimators of [1]} to more complex non-convex loss functions having heavy-tailed properties. To do so, we establish a set of sufficient conditions covering a large class of parametric gamma-frailty illness-death models, and verify the conditions for several common model specifications.
